A296792 Expansion of e.g.f. (sec(x) + tan(x))/sqrt(1 - 2*x).

1, 2, 6, 29, 196, 1721, 18622, 239427, 3563880, 60247537, 1139848346, 23857033243, 547234058732, 13650416199369, 367871731383990, 10651249531927427, 329733427896399952, 10868107639700229857, 379980639501713082034, 14046060369812427842859, 547335961798415004947220

Offset: 0

Author

Ilya Gutkovskiy, Dec 20 2017

Keywords

• nonn

Comments

Boustrophedon transform of A001147.

Links

• Robert Israel, Table of n, a(n) for n = 0..403
• J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996), 44-54 (Abstract, pdf, ps)
• N. J. A. Sloane, Transforms
• Index entries for sequences related to boustrophedon transform
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000111, A000754, A001147, A230960, A230961.

Programs

Maple

S:= series((sec(x)+tan(x))/sqrt(1-2*x), x, 51):
seq(coeff(S,x,n)*n!,n=0..50); # Robert Israel, Dec 21 2017

Mathematica

nmax = 20; CoefficientList[Series[(Sec[x] + Tan[x])/Sqrt[1 - 2 x], {x, 0, nmax}], x] Range[0, nmax]!

PARI

first(n) = x='x+O('x^n); Vec(serlaplace((1/cos(x) + tan(x))/sqrt(1-2*x))) \\ Iain Fox, Dec 21 2017

Python

from itertools import accumulate, count, islice
def A296792_gen(): # generator of terms
    blist, m = tuple(), 1
    for i in count(1,2):
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m *= i
A296792_list = list(islice(A296792_gen(),30)) # Chai Wah Wu, Jun 11 2022

Formula

a(n) ~ (sec(1/2) + tan(1/2)) * 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Dec 21 2017